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Let $G$ be a discrete group. Consider the category of $G$-simplicial sets endowed with the injective model structure, i.e. cofibrations are the injective maps and weak equivalences are the maps which are equivalences when seen as maps of simplicial sets.

My question is: what are the fibrations of this model category ? (aside from "the maps satisfying the right lifting property...")

This question of what are the fibrations of the injective model structure have already been asked several time on this forum and elsewhere, but it apparently never receive an answer in the more general case. I think this specific case is easier.

Indeed, I've been able to check that the trivial fibrations are exactly the maps $f$ such that for each subgroup $H \subset G$ the map $f$ restricted to a map between $H$-invariants is a trivial kan fibration.

It is hence natural to think that fibrations might be defined by the same property: for each $H$ the restriction of $f$ as a map between the $H$-invariants is a kan fibration.

It is indeed true that fibrations do satisfy this property, but as pointed out by Charles Rezk in the comments, this is not a sufficient condition.

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  • $\begingroup$ Wouldn't your description imply that the projective and injective model structures coincide? $\endgroup$
    – Matthias Wendt
    Commented Oct 17, 2014 at 12:32
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    $\begingroup$ There's a model structure on $G$-simplicial sets, where the cofibrations and fibrations are as you suggest, but the weak equivalences are equivariant ones (i.e., weak equivalence on $H$-invariants for all $H$). The fibrations in your injective structure will be characterized by more restrictive property. $\endgroup$
    – Charles Rezk
    Commented Oct 17, 2014 at 14:52
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    $\begingroup$ Matthias Wendt: No for the projective model structure the fibration and trivial fibration are defined by forgeting the G action, and the cofibration are inclusions $A \subset B$ where the elements of $B$ which are not in $A$ have a stabilizer reduced to $\{0\}$. Charles Rezk: thanks for that remark, this is really helpful ! (If someone can point a reference for this it might also be helpful) $\endgroup$
    – Simon Henry
    Commented Oct 18, 2014 at 8:05
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    $\begingroup$ @SimonHenry The model structure is the "usual" equivariant model structure. There are surely many references: maybe Dwyer & Kan, "Singular functors and realization functors". www3.nd.edu/~wgd/Dvi/SingularAndRealization.pdf $\endgroup$
    – Charles Rezk
    Commented Oct 18, 2014 at 15:44
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    $\begingroup$ The mixed model structure, whose fibrations and acyclic fibrations are as described in the main post, is explored by Shipley, see Proposition 1.3 in “A convenient model category for commutative ring spectra”. $\endgroup$
    – Dmitri Pavlov
    Commented Oct 19, 2014 at 11:42

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